Archive | 2021
On generators of the unstable $\\mathscr A$-module $H^{*}((K(\\mathbb F_2, 1))^{\\times 5})$ in a generic degree and applications
Abstract
Let us consider the prime field of two elements, $\\mathbb F_2.$ It is well-known that the classical hit problem for a module over the mod 2 Steenrod algebra $\\mathscr A$ is an interesting and important open problem of Algebraic topology, which asks a minimal set of generators for the polynomial algebra $\\mathcal P_m:=\\mathbb F_2[x_1, x_2, \\ldots, x_m]$, regarded as a connected unstable $\\mathscr A$-module on $m$ variables $x_1, \\ldots, x_m,$ each of degree 1. The algebra $\\mathcal P_m$ is the $\\mathbb F_2$-cohomology of the product of $m$ copies of the Eilenberg-MacLan complex $K(\\mathbb F_2, 1).$ Although the hit problem has been thoroughly studied for more than 3 decades, solving it remains a mystery for $m\\geq 5.$ The aim of this work is of studying the hit problem of five variables. More precisely, we develop our previous work \\cite{D.P3} on the hit problem for $\\mathscr A$-module $\\mathcal P_5$ in a degree of the generic form $n_t:=5(2^t-1) + 18.2^t,$ for any non-negative integer $t.$ An efficient approach to solve this problem had been presented. Moreover, we provide an algorithm in MAGMA for verifying the results and studying the hit problem in general. As an consequence, the calculations confirmed Sum s conjecture \\cite{N.S2} for the relationship between the minimal sets of $\\mathscr A$-generators of the polynomial algebras $\\mathcal P_{m-1}$ and $\\mathcal P_{m}$ in the case $m=5$ and degree $n_t.$ Two applications of this study are to determine the dimension of $\\mathcal P_6$ in the generic degree $5(2^{t+4}-1) + n_1.2^{t+4}$ for all $t > 0$ and to describe the modular representations of the general linear group of rank 5 over $\\mathbb F_2.$ As a corollary, the cohomological transfer , defined by W. Singer \\cite{W.S1}, is an isomorphism at the bidegree $(5, 5+n_0).$ Singer s transfer is one of the relatively efficient tools to approach the structure of mod-2 cohomology of the Steenrod algebra.